Bour's theorem for helicoidal surfaces with singularities

Abstract

In this paper, by generalizing the techniques of Bour's theorem, we prove that every generic cuspidal edge and, more generally, every generic n-type edge, which is invariant under a helicoidal motion in Euclidean 3-space admits non-trivial isometric deformations. As a corollary, several geometric invariants, such as the limiting normal curvature, the cusp-directional torsion, the higher order cuspidal curvature and the bias, are proved to be extrinsic invariants.